How to think of a set?

Solution 1:

The bad news

You're not alone in being confused about how to think of a set. Working out a sensible language for talking about sets cost mathematicians a great deal of sweat and tears. It's now considered one of the major achievements of 20th-century mathematics, and it's still being built upon today.

If you set out to explore our universe of sets, or the universes just beyond it, you'll quickly encounter sets that are dizzyingly, terrifyingly, hilariously counterintuitive. If you want my opinion, the best way to think of sets like these is don't.

The good news

On the other hand, if all you want to do is geometry, all the sets you meet will be tame enough that you can learn how to handle them just by looking at examples. In that spirit, let me give you some examples of sets that show up in geometry. It's often useful to have lots of examples, so this answer will be rather unfortunately long. Sorry!

Any finite collection of things. The three musketeers are a finite collection of people, so intuitively they should form a set, which I'll call $\mathbf{M}$. In formal notation, we might define this set by writing $$\mathbf{M} = \{\text{Athos}, \text{Porthos}, \text{Aramis}\}.$$ The individual musketeers are called the elements of this set. Where a normal person would say "Porthos is one of the three musketeers," a mathematician might say, "Porthos is an element of $\mathbf{M}$," or write $\text{Porthos} \in \mathbf{M}$.

The natural numbers. The numbers 1, 2, 3, 4, and so on are a collection of things, so intuitively they should form a set, which is usually called $\mathbb{N}$. Logicians, who need to use very formal language, talk about this set by giving a precise logical description of it. Fortunately, as a geometer, you can get away with an informal description like this: $$\mathbb{N} = \{1, 2, 3, 4, \ldots\}.$$ The dots are just another way of saying "and so on," in the hope that your reader will know what you mean.

The natural numbers whose squares are less than 100. These numbers are a collection of things, so they form a set, which I'll call $\mathbf{S}$. We can produce this set by rummaging through the set $\mathbb{N}$ and grabbing all the elements whose squares are less than 100. This way of building sets is so useful that people made up a special way of writing it: $$\mathbf{S} = \{n \in \mathbb{N} \mid n^2 < 100\}.$$ This could be read out loud as, "$\mathbf{S}$ is the set of natural numbers $n$ with the property that $n^2 < 100$."

Because every element of $\mathbf{S}$ is also an element of $\mathbb{N}$, we say $\mathbf{S}$ is a subset of $\mathbb{N}$. In writing, $\mathbf{S} \subset \mathbb{N}$.

If you do some thinking, you can find a more concrete expression for the set $\mathbf{S}$: $$\mathbf{S} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}.$$

The Euclidean plane. Imagine an infinite sheet of paper, perfectly flat and unmarked. Although it seems like a funny way of thinking at first, you can think of this sheet of paper as an infinite collection of locations, called points. This allows you to talk about the sheet of paper as a set, which I'll call $\mathbf{E}$.

One of the most interesting things you can do in the set $\mathbf{E}$ is measure distances. Given any two elements $p$ and $q$ of $\mathbf{E}$, I'll write the distance between them as $d(p, q)$.

A circle in the plane. A circle can be described as a set of points in the plane that are all the same distance from a certain point $a$. The point $a$ is called the center of the circle, and the common distance is called the radius. The circle with center $a \in \mathbf{E}$ and radius $1$, which I'll call $C_1(a)$, can be described formally by writing $$C_1(a) = \{p \in \mathbf{E} \mid d(p, a) = 1\}.$$ Every circle is a subset of $\mathbf{E}$.

A line in the plane. Pick two different points in the plane—that is, let's pick $a, b \in \mathbf{E}$ with $a \neq b$. Let's say $V(a, b)$ is the set of points which are equal distaces from $a$ and $b$. That is, $$V(a, b) = \{p \in \mathbf{E} \mid d(p, a) = d(p, b)\}.$$ If you think about it, you should be able to convince yourself that $L$ is a straight line!

Notice that many different choices of points $a$ and $b$ give the same line $V(a, b)$.

A set of circles in the plane. Sets are things, so a collection of sets should form a set! Logicians have to be very careful about constructing sets of sets, but as a geometer, you shouldn't run into any problems.

Here's a description of the set of all circles with radius one: $$\{C_1(a) \mid a \in \mathbf{E}\}.$$ If we unpack the meaning of $C_a(1)$, our description of this set of circles expands to $$\{\{p \in \mathbf{E} \mid d(p, a) = 1\} \mid a \in \mathbf{E}\}.$$ This expanded description makes it clear that we're dealing with a set of sets.

Another set of circles in the plane. Here's a description of the set of all circles whose radii are natural numbers: $$\{C_n(a) \mid a \in \mathbf{E}, n \in \mathbb{N}\}.$$

Every circle with radius one is a circle whose radius is a natural number, so $$\{C_1(a) \mid a \in \mathbf{E}\} \subset \{C_n(a) \mid a \in \mathbf{E}, n \in \mathbb{N}\}.$$

The set of all lines in the plane. Let's say $\mathcal{L}_{\mathbf{E}}$ is the set of all lines in the plane. This set can be described by writing $$\mathcal{L}_{\mathbf{E}} = \{V(a, b) \mid a, b \in \mathbf{E}\}.$$ If we actually went through all pairs of points $a, b \in \mathbf{E}$ and drew the line $V(a, b)$ for each pair, we would draw each line many times over. That's okay: when you're describing a set using this notation, it's okay if you describe some elements more than once.

The set $\mathcal{L}_{\mathbf{E}}$ is useful because it tells you a lot about the geometry of the plane. In fact, even if you forget how to measure distances, you can do a lot of geometry just by playing with the sets $\mathbf{E}$ and $\mathcal{L}_{\mathbf{E}}$. For example, here's an expression for the set of lines that pass through a pair of points $a$ and $b$: $$\{L \in \mathcal{L}_{\mathbf{E}} \mid a \in L \text{ and } b \in L\}.$$ It's a very important fact about geometry that when $a \neq b$, this set always has exactly one element!

The Fano plane. Once mathematicians got the hang of doing geometry by thinking of the plane as a set of points and then thinking of lines as a set of subsets of the plane, they realized that they might be able to create new kinds of geometry by starting with a set other than $\mathbf{E}$.

A cool example is the Fano plane, a set of seven "points" organized into "lines" of three points each. The relationship between the set of points $\mathbf{F}$ and the set of lines $\mathcal{L}_{\mathbf{F}}$ is very similar to the relationship between $\mathbf{E}$ and $\mathcal{L}_{\mathbf{E}}$. For instance, if you pick two points $a, b \in \mathbf{F}$ with $a \neq b$, you'll find that the set $$\{L \in \mathcal{L}_{\mathbf{F}} \mid a \in L \text{ and } b \in L\}$$ always has exactly one element. In the Fano plane, just like in the Euclidean plane, there's exactly one straight line through every two different points.

Solution 2:

Unfortunately, "set" is as nebulous a concept as sets themselves. It is not a geometric object, and certainly not a physical object. It is instead an abstract concept that arose from the desire to specify what kind of collections can be described. Clearly the specification would be essentially specifying some kind of property that the elements in a collection must satisfy and that everything else does not satisfy. At the same time, certain mathematicians wanted the elements of a set themselves to be sets for whatever reasons that are clearly not related to the physical world. And so naive set theory was constructed, where it is allowed to create any set (of sets) that satisfy a property. Unfortunately, this is just an identification of sets with functions from sets to boolean (true or false), and anything of this sort is doomed to a contradiction in any reasonable formal system. Therefore Zermelo and Fraenkel came up with a set of axioms where the kind of set you are allowed to construct is severely restricted but there are other axioms make up for that. In this way the naive idea of having sets identified with their indicator functions is preserved, but at the cost of having some collections being proper classes instead of sets. There are other alternatives such as various type theories some of which more closely reflect the actual way we do mathematics than ZFC. But that is another topic.

And if you study first-order logic, you will realize that ZFC is a first-order theory where the domain is intended to consist only of sets, but that is when viewing it from the outside! Within the theory you cannot define the domain at all, and all you 'know' are the axioms of ZFC. Similarly, the basic 2-input predicate "$\in$" cannot be defined within ZFC, because you cannot express "$t \in u$" if you are not allowed to use any predicates at all! Thus if you do mathematics completely within ZFC both sets and "$\in$" will be undefinable. How then do we talk about ZFC itself? We would have to go outside it. If you notice, when studying logic itself we are using some kind of mathematics to state and prove theorems about logic, and that mathematics is usually ZFC!

This phenomenon is intrinsic to all languages, including formal languages like ZFC and natural languages like English, where some notions are not definable in the language. For example, one cannot understand the meaning of "if" without already understanding something equivalent to it. This is in fact related to why we need at least one inference rule in logic, usually modus ponens, because without rules we cannot do anything. But rules cannot be defined! Moreover, to use modus ponens we already need to understand "if", because the rule says "If you have derived formulas $α,α \rightarrow β$, then you can derive $β$.".